Vector BC= vector AC- vector AB

Vector BM= 1/2* (vector BP+ vector BC)

= 1/2*[ (vector AP- vector AB)+ (vector AC- vector AB)]

= 1/2* vector AP+ 1/2* vector AC- vector AB

=x* vector AP+y* vector AC+z* vector AB

So x = 1/2, y = 1/2, z =- 1.